begin
theorem Th2:
the
InternalRel of
(Necklace 4) = {[0,1],[1,0],[1,2],[2,1],[2,3],[3,2]}
definition
existence
ex b1 being Subset of fin_RelStr st
( ( for R being strict RelStr st the carrier of R is 1 -element & the carrier of R in FinSETS holds
R in b1 ) & ( for H1, H2 being strict RelStr st the carrier of H1 misses the carrier of H2 & H1 in b1 & H2 in b1 holds
( union_of (H1,H2) in b1 & sum_of (H1,H2) in b1 ) ) & ( for M being Subset of fin_RelStr st ( for R being strict RelStr st the carrier of R is 1 -element & the carrier of R in FinSETS holds
R in M ) & ( for H1, H2 being strict RelStr st the carrier of H1 misses the carrier of H2 & H1 in M & H2 in M holds
( union_of (H1,H2) in M & sum_of (H1,H2) in M ) ) holds
b1 c= M ) )
uniqueness
for b1, b2 being Subset of fin_RelStr st ( for R being strict RelStr st the carrier of R is 1 -element & the carrier of R in FinSETS holds
R in b1 ) & ( for H1, H2 being strict RelStr st the carrier of H1 misses the carrier of H2 & H1 in b1 & H2 in b1 holds
( union_of (H1,H2) in b1 & sum_of (H1,H2) in b1 ) ) & ( for M being Subset of fin_RelStr st ( for R being strict RelStr st the carrier of R is 1 -element & the carrier of R in FinSETS holds
R in M ) & ( for H1, H2 being strict RelStr st the carrier of H1 misses the carrier of H2 & H1 in M & H2 in M holds
( union_of (H1,H2) in M & sum_of (H1,H2) in M ) ) holds
b1 c= M ) & ( for R being strict RelStr st the carrier of R is 1 -element & the carrier of R in FinSETS holds
R in b2 ) & ( for H1, H2 being strict RelStr st the carrier of H1 misses the carrier of H2 & H1 in b2 & H2 in b2 holds
( union_of (H1,H2) in b2 & sum_of (H1,H2) in b2 ) ) & ( for M being Subset of fin_RelStr st ( for R being strict RelStr st the carrier of R is 1 -element & the carrier of R in FinSETS holds
R in M ) & ( for H1, H2 being strict RelStr st the carrier of H1 misses the carrier of H2 & H1 in M & H2 in M holds
( union_of (H1,H2) in M & sum_of (H1,H2) in M ) ) holds
b2 c= M ) holds
b1 = b2
end;
Lm1:
the carrier of (Necklace 4) = {0,1,2,3}
by NECKLACE:1, NECKLACE:20;